# Cubic reciprocity

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In mathematics, cubic reciprocity refers to various results connecting the solvability of two related cubic equations in modular arithmetic. It is a generalisation of the concept of quadratic reciprocity.

## Algebraic setting

The law of cubic reciprocity is most naturally expressed in terms of the Eisenstein integers, that is, the ring E of complex numbers of the form

${\displaystyle z=a+b\,\omega }$

where and a and b are integers and

${\displaystyle \omega ={\frac {1}{2}}(-1+i{\sqrt {3}})=e^{2\pi i/3}}$

is a complex cube root of unity.

If ${\displaystyle \pi }$ is a prime element of E of norm P and ${\displaystyle \alpha }$ is an element coprime to ${\displaystyle \pi }$, we define the cubic residue symbol ${\displaystyle \left({\frac {\alpha }{\pi }}\right)_{3}}$ to be the cube root of unity (power of ${\displaystyle \omega }$) satisfying

${\displaystyle \alpha ^{(P-1)/3}\equiv \left({\frac {\alpha }{\pi }}\right)_{3}\mod \pi }$

We further define a primary prime to be one which is congruent to -1 modulo 3. Then for distinct primary primes ${\displaystyle \pi }$ and ${\displaystyle \theta }$ the law of cubic reciprocity is simply

${\displaystyle \left({\frac {\pi }{\theta }}\right)_{3}=\left({\frac {\theta }{\pi }}\right)_{3}}$

with the supplementary laws for the units and for the prime ${\displaystyle 1-\omega }$ of norm 3 that if ${\displaystyle \pi =-1+3(m+n\omega )}$ then

${\displaystyle \left({\frac {\omega }{\pi }}\right)_{3}=\omega ^{m+n}}$
${\displaystyle \left({\frac {1-\omega }{\pi }}\right)_{3}=\omega ^{2m}}$

## References

• David A. Cox, Primes of the form ${\displaystyle x^{2}+ny^{2}}$, Wiley, 1989, ISBN 0-471-50654-0.
• K. Ireland and M. Rosen, A classical introduction to modern number theory, 2nd ed, Graduate Texts in Mathematics 84, Springer-Verlag, 1990.
• Franz Lemmermeyer, Reciprocity laws: From Euler to Eisenstein, Springer Verlag, 2000, ISBN 3-540-66957-4.